Limit Theorems and Statistical Inference for Ergodic Processes

0102268遍历过程的极限定理和统计推断该项目的一个主要目标是发展一种新的方法来解决变点问题，其中后者的突变被任意的单调变化所取代。新的方法使用惩罚似然比统计量来检验均值与非递减趋势的相等性，该趋势是针对独立的正常观测误差而得出的。测试的性质可以在更一般的依赖但平稳和遍历误差的背景下进行研究。在分析灾难性事件或法律干预导致的气候变化结果时，这种方法的应用应该是显而易见的，例如要求减少车辆排放。目前，主要研究人员和学生的工作已经确定了适度条件下平稳遍历误差的检验统计量的渐近零分布，从而允许应用于历史数据集，如天气数据。剩下的问题包括开发应用于质量控制的序贯模拟，替代处罚，以及在保序回归后估计方差参数。该项目的第二个主要目标是在适用于第一个的背景下发展渐近分布理论。研究马尔可夫链的可加泛函的中心极限定理，特别是当前状态是先前状态的函数和自变量的链。许多线性和非线性时间序列模型都是这种形式的。对于这类过程，平稳分布存在的条件已被广泛研究，但对其可加泛函的中心极限理论的研究工作要少得多。首席研究员计划在此背景下发展中心极限理论。前人的工作表明，在许多情况下，可加泛函可以写成一个鞅加上一个较低阶的余项，然后可以从鞅中心极限定理推导出渐近正态。这种方法不需要Harris递推或其他强形式的渐近独立性。它将被开发，并探索统计应用，特别是在修正的变点问题上的应用。其他统计应用包括设置近似置信度区间。在某些情况下，可以从多元中心极限定理获得近似的置信度区间。对于其他人来说，有必要发展经验过程的紧密性，这个问题将被研究。在高度结构化的模型中，可以超越渐近正态分布而得到(类似Edgeworth的)渐近展开，由此可以形成修正的置信度区间，这些区间的实际覆盖概率以较快的速度收敛到标称值。该项目的第三个主要目标是开发此类扩建项目。主要研究人员、同事和学生之前的工作已经为自适应设计的线性模型和自回归过程开发了这种性质的扩展。这项工作将被推广到有限维分布形成指数族的过程，这是一大类包括马尔可夫链和许多半马尔可夫过程的过程。

Abstract 0102268Limit Theorems and Statistical Inference for Ergodic ProcessesA major goal of the project is to develop a new approach to the change point problem in which the abrupt change of the latter is replaced by an arbitrary monotonic change. The new procedure uses a penalized likelihood ratio statistic for testing equality of means against a non-decreasing trend, derived for independent normal observation errors. The properties of the test can be studied in the more general context of dependent, but stationary and ergodic errors. Applications of such procedures should be evident in the analysis of climate changes results from cataclysmic events or legal intervention, such as the required reduction on vehicle emissions.Current work by the principal investigator and students has determined the asymptotic null distribution of the test statistic for stationary ergodic errors under modest conditions, thus allowing application to historical data sets, like weather data. Remaining questions include developing a sequential analogue for applications to quality control, alternative penalizations, and estimating a variance parameter after an isotonic regression. A second major goal of the project is to develop asymptotic distribution theory in a context that is applicable to the first. The central limit theorem will be studied for additive functionals of a Markov chain with special attention to chains in which the current state is a function of the previous state and an independent variable. Many linear and non-linear time series models are of this form. Conditions for the existence of a stationary distribution have been widely studied for such processes, but there is much less work on central limit theory for their additive functionals. The principal investigator plans to develop central limit theory in this context. Previous work has shown that additive functionals can be written as a martingale plus a remainder term of smaller order in many cases, and then asymptotic normality can be deduced from the martingale central limit theorem. This approach does not require Harris recurrence or other strong forms of asymptotic independence. It will be developed, and statistical applications explored, especially applications to the modified change point problem. Other statistical applications include setting approximate confidence intervals. In some cases, approximate confidence intervals may be obtained from a multivariate central limit theorem. For others, it is necessary to develop tightness of empirical processes, and this question will be studied. In highly structured models, it is possible to go beyond asymptotic normality to (Edgeworth like) asymptotic expansions from which corrected confidence intervals can be formed, intervals whose actual coverage probability converges to the nominal value at a fast rate. A third major objective of project is to develop such expansions. Previous work by the principal investigator, co-workers, and students has developed expansions of this nature for adaptively designed linear models and auto regressive processes. This work will be extended to processes whose finite dimensional distributions form exponential families, a large class of processes that includes Markov Chains and many semi-Markov processes.